### Abstract

We present a new proof rule for proving almost-sure termination of probabilistic programs, including those that contain demonic non-determinism.

An important question for a probabilistic program is whether the probability mass of all its diverging runs is zero, that is that it terminates "almost surely". Proving that can be hard, and this paper presents a new method for doing so. It applies directly to the program's source code, even if the program contains demonic choice.

Like others, we use variant functions (a.k.a. "super-martingales") that are real-valued and decrease randomly on each loop iteration; but our key innovation is that the amount as well as the probability of the decrease are parametric. We prove the soundness of the new rule, indicate where its applicability goes beyond existing rules, and explain its connection to classical results on denumerable (non-demonic) Markov chains.

An important question for a probabilistic program is whether the probability mass of all its diverging runs is zero, that is that it terminates "almost surely". Proving that can be hard, and this paper presents a new method for doing so. It applies directly to the program's source code, even if the program contains demonic choice.

Like others, we use variant functions (a.k.a. "super-martingales") that are real-valued and decrease randomly on each loop iteration; but our key innovation is that the amount as well as the probability of the decrease are parametric. We prove the soundness of the new rule, indicate where its applicability goes beyond existing rules, and explain its connection to classical results on denumerable (non-demonic) Markov chains.

Original language | English |
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Article number | 33 |

Pages (from-to) | 1-28 |

Number of pages | 28 |

Journal | Proceedings of the ACM on Programming Languages |

Volume | 2 |

Issue number | POPL |

DOIs | |

Publication status | Published - 5 Jan 2018 |

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### Keywords

- Almost-sure termination
- demonic non-determinism
- program logic pGCL

### Cite this

McIver, A., Morgan, C., Kaminski, B. L., & Katoen, J-P. (2018). A new proof rule for almost-sure termination.

*Proceedings of the ACM on Programming Languages*,*2*(POPL), 1-28. [33]. https://doi.org/10.1145/3158121